#7712788

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,12,16)(7,14,10,9)(8,15,11,13)(17,20,18,19)(21,22,26,24)(23,27,31,29)(25,32,33,28)(30,35,36,34);(1,2,3,4)(5,7,12,10)(6,9,16,14)(8,18,11,17)(13,19,15,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,8,12,11)(6,13,16,15)(7,17,10,18)(9,20,14,19)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(5,9,12,14)(6,7,16,10)(8,19,11,20)(13,18,15,17);(5,10,12,7)(6,14,16,9)(8,17,11,18)(13,20,15,19)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,11,12,8)(6,15,16,13)(7,17,10,18)(9,20,14,19)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619984);